Everyone have some experience how radio turns into static when one is driving into a tunnel. It just does not work (unless there is a special radiating cable installed inside the tunnel).
Once upon a night I could not get sleep pondering on why does this happen. For example, water seems to behave quite differently; no matter how narrow the pipe, a water wave will go through it. Someone might have told me that FM radio wavelength is just so large that the wave “does not fit the tunnel”. But as intuitively acceptable this explanation could be (I don't think it is even that), it is not a very satisfying explanation as such.
After some research I found that the explanation lies in the behavior of Maxwell equations (well, what a surprise). All backs down to the boundary conditions of the electromagnetic field at the tunnel wall: the tangential component of the electrical field component of the electromagnetic wave has to be near to zero in the tunnel interface. This leads to a situation where the wave has, in a sense, no room the oscillate inside the tunnel. The zero interface condition here is crucial; for example, it does not generally apply to a water wave and this is the reason a wave in the water fit through the smallest hole.
Intuitive or not, this is what the equations tells us. To demonstrate this fact I solved, using FEM, the Helmholtz equation for an plane-wave in a two dimensional setting mimicking a tunnel and its entrance. The electromagnetic wave is considered to be polarized so that the electrical component is pointing at right angle w.r.t. the plane (either towards or against the reader of this page). Boundary conditions inside the 2D tunnel is set to zero. Solutions coloring shown in the figures following represents the magnitude of the electrical component.
We compare the behavior of two different wavelengths: